cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A207969 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^4 * x^n/n ).

Original entry on oeis.org

1, 5, 15, 60, 295, 1625, 9430, 56465, 345010, 2139595, 13419500, 84926105, 541398665, 3472389210, 22385362895, 144945232375, 942089445030, 6143582084115, 40181143112035, 263482860974570, 1731780213622125, 11406235045261205, 75268685723935940
Offset: 0

Views

Author

Paul D. Hanna, Feb 22 2012

Keywords

Comments

Conjecture: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k) * x^n/n ) is an integer series for integers k>=0.
Note that exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k+1) * x^n/n ) is not an integer series for integers k.
Note that exp( Sum_{n>=1} Fibonacci(n)^(2*k) * x^n/n ) is not an integer series for integers k.

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + 295*x^4 + 1625*x^5 + 9430*x^6 +...
such that
log(A(x))/5 = x + x^2/2 + 2^4*x^3/3 + 3^4*x^4/4 + 5^4*x^5/5 + 8^4*x^6/6 + 13^4*x^7/7 +...+ Fibonacci(n)^4*x^n/n +...

Crossrefs

Cf. A054888, A207970, A207971, A207972, A207834, A207835.
A077916, A203804.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,5*fibonacci(k)^4*x^k/k)+x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

Formula

The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + ... is an algebraic function: A(x)^5 = (1 + 3*x + x^2)^4/( (1 - 7*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A203804. - Peter Bala, Apr 03 2014
a(n) ~ 2^(4/5) * 5^(1/10) * phi^(4*n) / (Gamma(1/5) * 3^(1/5) * n^(4/5)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 18 2020

A207834 G.f.: exp( Sum_{n>=1} 5*L(n)*x^n/n ), where L(n) = Fibonacci(n-1)^n + Fibonacci(n+1)^n.

Original entry on oeis.org

1, 5, 25, 130, 1295, 38861, 4227075, 1309117220, 1123176929475, 2564594183278115, 15604715134340991949, 251021373648740285348860, 10668788238489683954523431475, 1195322752666989652479885363067075, 352750492054485236937115646128341734205
Offset: 0

Views

Author

Paul D. Hanna, Feb 20 2012

Keywords

Comments

Given g.f. A(x), note that A(x)^(1/5) is not an integer series.
Compare the definition to the g.f. of the Fibonacci numbers:
1/(1-x-x^2) = exp( Sum_{n>=1} Lucas(n)*x^n/n ), where Lucas(n) = Fibonacci(n-1) + Fibonacci(n+1).

Examples

			G.f.: A(x) = 1 + 5*x + 25*x^2 + 130*x^3 + 1295*x^4 + 38861*x^5 +...
such that, by definition,
log(A(x))/5 = x + 5*x^2/2 + 28*x^3/3 + 641*x^4/4 + 33011*x^5/5 +...+ (Fibonacci(n-1)^n + Fibonacci(n+1)^n)*x^n/n +...

Crossrefs

Cf. A207835, A156216, A166168.

Programs

  • PARI
    {L(n)=fibonacci(n-1)^n+fibonacci(n+1)^n}
{a(n)=polcoeff(exp(sum(m=1,n,5*L(m)*x^m/m)+x*O(x^n)),n)}
for(n=0,51,print1(a(n),", "))

A207384 A206815(n+1)-A206815(n).

Original entry on oeis.org

1, 3, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 1, 2, 3, 2, 2, 1, 3, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2012

Keywords

Comments

The sequences A206815, A206818, A207384, A207835 illustrate the closeness of {j+pi(j)} to {k+(k+1)/log(k+1)}, as suggested by the prime number theorem and the conjecture that all the terms of A207384 and A207835 are in the set {1,2,3}.

Examples

			The joint ranking is represented by
1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...
Positions of numbers j+pi(j): 1,2,5,7,9,...
Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..

Crossrefs

Cf. A000720, A206815, A206818.

Programs

  • Mathematica
    f[1, n_] := n + PrimePi[n];
f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)
Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)
d1[n_] := p[1, n + 1] - p[1, n]
Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A207385 *)
d2[n_] := p[2, n + 1] - p[2, n]
Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A207386 *)

A207385 A206818(n+1)-A206818(n).

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 2, 2, 1, 3, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 3, 1, 2, 2, 1, 2, 3, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 1
Offset: 1

Views

Author

Clark Kimberling, Feb 17 2012

Keywords

Comments

The sequences A206815, A206818, A207384, A207835 illustrate the closeness of {j+pi(j)} to {k+(k+1)/log(k+1)}, as suggested by the prime number theorem and the conjecture that all the terms of A207384 and A207835 are in the set {1,2,3}.

Examples

			The joint ranking is represented by
1 < 3 < 3.8 < 4.7 < 5 < 5.8 < 6 <7.1 < 8 < 8.3 < 9 < ...
Positions of numbers j+pi(j): 1,2,5,7,9,...
Positions of numbers k+(k+1)/log(k+1): 3,4,6,8,10,..

Crossrefs

Cf. A000720, A206815, A206818.

Programs

  • Mathematica
    f[1, n_] := n + PrimePi[n];
f[2, n_] := n + N[(n + 1)/Log[n + 1]]; z = 500;
t[k_] := Table[f[k, n], {n, 1, z}];
t = Sort[Union[t[1], t[2]]];
p[k_, n_] := Position[t, f[k, n]];
Flatten[Table[p[1, n], {n, 1, z}]]    (* A206815 *)
Flatten[Table[p[2, n], {n, 1, z}]]    (* A206818 *)
d1[n_] := p[1, n + 1] - p[1, n]
Flatten[Table[d1[n], {n, 1, z - 1}]]  (* A207385 *)
d2[n_] := p[2, n + 1] - p[2, n]
Flatten[Table[d2[n], {n, 1, z - 1}]]  (* A207386 *)
Showing 1-4 of 4 results.

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